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High-Order Implicit Low-Rank Method with Spectral Deferred Correction for Matrix Differential Equations

Shun Li, Yan Jiang, Yingda Cheng

Abstract

In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. In [1], a low rank numerical method is proposed to correct the modeling error of the basis update and the Galerkin (BUG) method, which is a computational approach for DLRA. This method (merge-BUG/mBUG method) has been demonstrated to be first order convergent for general advection-diffusion problems. In this paper, we explore using SDC to elevate the convergence order of the mBUG method. In SDC, we start by computing a first-order solution by mBUG, and then perform successive updates by computing low-rank solutions to the Picard integral equation. Rather than a straightforward application of SDC with mBUG, we propose two aspects to improve computational efficiency. The first is to reduce the intermediate numerical rank by detailed analysis of dependence of truncation parameter on the correction levels. The second aspect is a careful choice of subspaces in the successive correction to avoid inverting large linear systems (from the K- and L-steps in BUG). We prove that the resulting scheme is high-order accurate for the Lipschitz continuous and bounded dynamical system. We consider numerical rank control in our framework by comparing two low-rank truncation strategies: the hard truncation strategy by truncated singular value decomposition and the soft truncation strategy by soft thresholding. We demonstrate numerically that soft thresholding offers better rank control in particular for higher-order schemes for weakly (or non-)dissipative problems.

High-Order Implicit Low-Rank Method with Spectral Deferred Correction for Matrix Differential Equations

Abstract

In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. In [1], a low rank numerical method is proposed to correct the modeling error of the basis update and the Galerkin (BUG) method, which is a computational approach for DLRA. This method (merge-BUG/mBUG method) has been demonstrated to be first order convergent for general advection-diffusion problems. In this paper, we explore using SDC to elevate the convergence order of the mBUG method. In SDC, we start by computing a first-order solution by mBUG, and then perform successive updates by computing low-rank solutions to the Picard integral equation. Rather than a straightforward application of SDC with mBUG, we propose two aspects to improve computational efficiency. The first is to reduce the intermediate numerical rank by detailed analysis of dependence of truncation parameter on the correction levels. The second aspect is a careful choice of subspaces in the successive correction to avoid inverting large linear systems (from the K- and L-steps in BUG). We prove that the resulting scheme is high-order accurate for the Lipschitz continuous and bounded dynamical system. We consider numerical rank control in our framework by comparing two low-rank truncation strategies: the hard truncation strategy by truncated singular value decomposition and the soft truncation strategy by soft thresholding. We demonstrate numerically that soft thresholding offers better rank control in particular for higher-order schemes for weakly (or non-)dissipative problems.

Paper Structure

This paper contains 10 sections, 3 theorems, 68 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background review
  3. Notations and preliminaries
  4. The mBUG method
  5. Spectral deferred correction method
  6. Numerical method
  7. The SDC-mBUG method
  8. Theoretical analysis
  9. Numerical results
  10. Conclusions and future work

Key Result

Theorem 2.1

Suppose that $X_n = X(t_n)$, the local truncation error of the SDC scheme is where $h = \max_{m}\{\Delta t_{n,m}\}$.

Figures (4)

  • Figure 1: Example \ref{['newEg1']}. The rank evolution of the numerical solutions obtained by the SDC-mBUG schemes over time is shown, where the dashed line represents the rank evolution of the reference solution. Top: hard truncation; bottom: soft truncation. Left: the third-order SDC-mBUG scheme; right: the fourth-order SDC-mBUG scheme.
  • Figure 2: Example \ref{['newEg3']}. The rank evolution of the numerical solutions obtained by the SDC-mBUG schemes over time is shown, where the dashed line represents the rank evolution of the reference solution. Top: hard truncation; bottom: soft truncation. Left: the third-order SDC-mBUG scheme; right: the fourth-order SDC-mBUG scheme.
  • Figure 3: Example \ref{['newEg4']}. The rank evolution of the numerical solutions obtained by the SDC-mBUG schemes over time is shown, where the dashed line represents the rank evolution of the reference solution. Top: hard truncation; bottom: soft truncation. Left: the third-order SDC-mBUG scheme; right: the fourth-order SDC-mBUG scheme.
  • Figure 4: Example \ref{['newEg5']}. The rank evolution of the numerical solutions obtained by the SDC-mBUG schemes over time is shown, where the dashed line represents the rank evolution of the reference solution. Top: hard truncation; bottom: soft truncation. Left: the third-order SDC-mBUG scheme; right: the fourth-order SDC-mBUG scheme.

Theorems & Definitions (10)

  • Theorem 2.1
  • Remark
  • Lemma 3.1: Local truncation error of first order mBUG scheme appelo2024robust
  • Theorem 3.2: Local truncation error of SDC-mBUG scheme
  • proof
  • Remark 3.1
  • Example 1: Manufactured solution
  • Example 2: Schrödinger equation
  • Example 3: Rotation with anisotropic diffusion
  • Example 4: Pure rotation